We study the doubly nonlinear PDE vertical bar partial derivative u vertical bar(p-2)partial derivative(t)u - div(vertical bar del u vertical bar(p-2)del u) = 0. This equation arises in the study of extremals of Poincare inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Holder continuity in time of order (p - 1)/p for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.